Carnival of Mathematics #60

Welcome to the Carnival of Mathematics! Finding that the 60th is apparently the “diamond anniversary,” I was reminded of the symmetry in the Buckyball, which has the shape of a truncated icosahedron. You can make pretty nice ones using modular origami:

Before getting to this month’s links, allow me a diversion to talk about some geometry I learned a little of this month.

There are 6!=720 ways to order the letters A, B, E, I, L, and S. If we declare that two orderings are the same if one is obtained from the other by cyclic permutation (for example, ABEILS and ILSABE are the same), there are 6!/6=5!=120 combinations. If we also declare that a word and it’s reverse are the same (ABEILS = SLIEBA), we have arrived at 6!/(6*2)=60 combinations.

Pick any 6 distinct points on a circle (or any conic section). Choose any of the points as a starting point, and draw a line to any of the other points. Then draw a line to one of the remaining 4 points. Continue until all of the points have been hit, and then draw a line back to your starting point. How many different pictures can you make in this process? 60, again, because you could label the points A, B, E, I, L, S, and then pictures correspond to words from the previous calculation.

Each picture you draw is a figure with six edges. These six edges can be put into three set of pairs, where two edges are paired if they are “opposite.” In the process of drawing the lines, above, the line opposite the very first line is the fourth line you draw. Similarly, the second and fifth form a pair, and then the third and sixth.

Now, if you extend all of the lines, each pair of opposite edges will determine a point of intersection (or infinity… maybe try another setup for your original points :)). So each picture you draw determines 3 points in the plane (or infinity). When he was only 16, Pascalshowed that these three points are always colinear.

So, given 6 points on a conic, the process outlined above determines 60 lines, called Pascal Lines. Mathworld has more on Pascal Lines, for the inquisitive, so it’s probably about time to direct you over there and get on to this month’s blog posts!

2) At approximately halfway through the month, Wednesday the 18th was determined to be the 150th birthday of the Riemann Hypothesis. Plus Magazine and Math In The News both had articles.

3) Riemann’s zeta function, the lead character in his hypothesis, is connected to primes by Euler’s product formula. If you are interested in the distribution of the primes, Matt Springer at Built on Facts has a post about the function Li(x), as part of his running Sunday Function series. If natural number primes aren’t exciting enough for you, Rich Beveridge at Where The Arts Meet The Sciences has a post for you on Gaussian Primes.

If you prefer unsolved puzzles, Bill the Lizard has recently written posts about the Collatz Conjecture and the Perfect Cuboid Problem. Alternatively, for some behind-the-scenes on the notoriously difficult Putnam exam (and yet more puzzles), head over to Izabella’s post at The Accidental Mathematician.

5) It’ll take a while to get to the 3435th Carnival of Math, so I think I’m not stepping on too many toes if I point you at Mike Croucher’s quick post at Walking Randomly and Dan MacKinnon’s slightly longer post at mathrecreation that talk about what makes 3435 interesting.

30) Dave Richeson at Division By Zero reports on a case of mistaken identity in Legendre Who?

60) Finally, Samuel at ACME Science discusses the fractured state of the current mathematics community, noting that Mathematics Really is Discrete. This post was closely followed by Abstruse Goose’s Landscape.